Yes the meeting was in breach of the rules because meetings of Parish Councils must be held in public and proper notice (the 3 days) must have been given to the public. The council can vote to exclude the public for a particular item on the agenda if there are grounds to do so. This would usually be if the item is of a confidential nature such as the appointment of a clerk or review of the clerk's terms of employment, as an example. By logical extension, the council may resolve to exclude the public from an item and then resolve to adjourn the meeting to a later date to siscuss this particular business. In such circumstances, further notice of the adjournment is not required because the notice requirements are covered by the original notice. From what you say, your chair (the vice-chairman) simply called and held a meeting in private without notice. That is against the rules.
There is no question that the Vice-chairman cannot chair meetings following the departure of the chair although the appointment of an actual chair should be a priority.